Related papers: Consecutive Radio Labeling of Hamming Graphs

The famous K\H{o}nig-Egerv\'ary theorem is equivalent to the statement that the matching number equals the vertex cover number for every induced subgraph of some graph if and only if that graph is bipartite. Inspired by this result, we…

An {\it additive labeling} of a graph $G$ is a function $ \ell :V(G) \rightarrow\mathbb{N}$, such that for every two adjacent vertices $ v $ and $ u$ of $ G $, $ \sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w) $ ($ x \sim y $ means that $…

A $k$-ranking is a vertex $k$-coloring such that if two vertices have the same color any path connecting them contains a vertex of larger color. The rank number of a graph is smallest $k$ such that $G$ has a $k$-ranking. For certain graphs…

Structured prediction tasks in machine learning involve the simultaneous prediction of multiple labels. This is typically done by maximizing a score function on the space of labels, which decomposes as a sum of pairwise elements, each…

Which graphs admit an integer value harmonic function which is injective and surjective onto $\Z$? Such a function, which we call harmonic labeling, is constructed when the graph is the $\Z^2$ square grid. It is shown that for any finite…

A mapping $\alpha : V(G) \to V(H)$ from the vertex set of one graph $G$ to another graph $H$ is an isometric embedding if the shortest path distance between any two vertices in $G$ equals the distance between their images in $H$. Here, we…

An automorphism on a graph $G$ is a bijective mapping on the vertex set $V(G)$, which preserves the relation of adjacency between any two vertices of $G$. An automorphism $g$ fixes a vertex $v$ if $g$ maps $v$ onto itself. The stabilizer of…

A graph $G$ is perfectly divisible if, for every induced subgraph $H$ of $G$, either $V(H)$ is a stable set or admits a partition into two sets $X_1$ and $X_2$ such that $\omega(H[X_1]) < \omega(H)$ and $H[X_2]$ is a perfect graph. In this…

We present a compact labeling scheme for determining whether a designated set of terminals in a graph remains connected after any $f$ (or less) vertex failures occur. An $f$-FT Steiner connectivity labeling scheme for an $n$-vertex graph…

A set $B$ of vertices in a graph $G$ is called a \emph{$k$-limited packing} if for each vertex $v$ of $G$, its closed neighbourhood has at most $k$ vertices in $B$. The \emph{$k$-limited packing number} of a graph $G$, denoted by $L_k(G)$,…

A successive vertex ordering of a graph is a linear ordering of its vertices in which every vertex except the first has at least one neighbour appearing earlier. Such orderings arise naturally in incremental growth and…

A connected graph $G$ with at least two vertices is matching covered if each of its edges lies in a perfect matching. A matching covered graph is minimal if the removal of any edge results in a graph that is no longer matching covered. An…

An $L(2,1)$-labeling of a graph $G=(V,E)$ is a function $f$ from the vertex set $V(G)$ to the set of nonnegative integers such that the labels on adjacent vertices differ by at least two, and the labels on vertices at distance two differ by…

For an ordered set $W=\{w_1,w_2,...,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the ordered $k$-vector $r(v|W):=(d(v,w_1),d(v,w_2),...,d(v,w_k))$ is called the (metric) representation of $v$ with respect to $W$, where…

Given positive integers $k$ and $\ell$ we write $G \rightarrow (K_k,K_\ell)$ if every 2-colouring of the edges of $G$ yields a red copy of $K_k$ or a blue copy of $K_\ell$ and we denote by $R(k)$ the minimum $n$ such that $K_n\rightarrow…

For integer $k\geq2,$ a graph $G$ is called $k$-leaf-connected if $|V(G)|\geq k+1$ and given any subset $S\subseteq V(G)$ with $|S|=k,$ $G$ always has a spanning tree $T$ such that $S$ is precisely the set of leaves of $T.$ Thus a graph is…

Given an edge labeling $f$ of a graph $G$, a vertex $v$ is called an $AR$-vertex, if $v$ has distinct edge weight sums for each distinct subset of edges incident on $v$. An injective edge labeling $f$ of a graph $G$ is called an…

Let $G=(V,E)$ be a simple graph with maximum degree $d$. For an integer $k\in\mathbb{N}$, the $k$-disc of a vertex $v\in V$ is defined as the rooted subgraph of $G$ that is induced by all vertices whose distance to $v$ is at most $k$. The…

Random graphs are an important tool for modelling and analyzing the underlying properties of complex real-world networks. In this paper, we study a class of random graphs known as the inhomogeneous random K-out graphs which were recently…

A complete $k$-coloring of a graph $G=(V,E)$ is an assignment $\varphi:V\to\{1,\ldots,k\}$ of colors to the vertices such that no two vertices of the same color are adjacent, and the union of any two color classes contains at least one…